Critical behaviours of the fifth Painlev\'e transcendents and the monodromy data

TL;DR

This paper analyzes the critical behaviors of fifth Painlevé transcendents, presenting convergent series solutions, monodromy data, and the structure of their analytic continuation, zeros, poles, and 1-points.

Contribution

It introduces new families of solutions near the origin and details their monodromy data, including non-generic cases and transformations, advancing understanding of Painlevé V transcendents.

Findings

Convergent series solutions near the origin are constructed.

Monodromy data are computed for various solution types.

The structure of zeros, poles, and 1-points is clarified.

Abstract

For the fifth Painlev\'e equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. It is also possible to compute the monodromy data in non-generic cases. Solutions of logarithmic type are derived from those of inverse logarithmic type through a B\"{a}cklund transformation found by Gromak. In a special case the complex power type of solutions have relatively simple oscillatory expressions. For the complex power type of solutions in the generic case, we clarify the structure of the analytic continuation on the universal covering around the origin, and examine the distribution of zeros, poles and 1-points. It is shown that two kinds of spiral domains including a sector as a special case are alternately arrayed; the domains of one kind contain sequences both of zeros and of poles, and those of the other kind sequences of 1-points.